Problem 86 - College Algebra Essentials

Question

Find all values of \(x\) satisfying the given conditions. $$y=|2-3 x| \text { and } y=13$$

Step-by-Step Solution

Answer

The solutions to the given conditions are \(x = \frac{-11}{3}\) and \(x = 5\).

Step 1: Set Up Two Equations

Since an absolute value equation can either be positive or negative, we setup two equations to solve for x: 2-3x = 13 and 2-3x = -13.

Step 2: Solve the First Equation

Solve the equation 2-3x = 13 for x. Subtract 2 from both sides to get -3x = 11. Then, divide each side by -3 to solve for x. This gives us \(x = \frac{-11}{3}\).

Step 3: Solve the Second Equation

Solve the equation 2-3x = -13 for x. Subtract 2 from both sides of the equation to get -3x = -15. Dividing both sides by -3 yields \(x = \frac{15}{3}\), which simplifies to \(x = 5\).

Key Concepts

Solving Linear Equations

When tasked with solving linear equations involving absolute values, breaking them down into simpler steps is key. In the given exercise, we start with the equation \(y = |2 - 3x|\) and \(y = 13\). The absolute value symbol means the expression inside can be both a positive or a negative quantity with the same absolute magnitude.
To find solutions for \(x\), we first set up two separate linear equations based on the absolute value properties:

• The expression inside the absolute value equals the positive case: \(2 - 3x = 13\)
• The expression inside equals the negative case: \(2 - 3x = -13\)

By solving these equations separately, we determine the potential values of \(x\) that satisfy the conditions outlined by the original problem. After isolating \(x\) in each equation, we're able to find the solutions that make both sides of the equation equal, reflecting both scenarios.

Algebraic Expressions

Algebraic expressions like \(2 - 3x\) play a crucial role in solving equations. They involve variables (like \(x\)), constants, and arithmetic operations.
In the exercise, the expression \(2 - 3x\) is manipulated using arithmetic operations to find solutions for \(x\):

• In the first equation, \(2 - 3x = 13\): Subtraction is used to isolate the \(-3x\) term, followed by division to solve for \(x\).
• For the second equation, \(2 - 3x = -13\): Similar manipulations are applied—subtraction followed by division.

The goal is to simplify the equation step by step until we are left with \(x\) isolated on one side. Understanding how to manipulate algebraic expressions is vital for unraveling more complex mathematical problems.

Mathematical Reasoning

Mathematical reasoning involves logically approaching problems to find solutions. It is about understanding the rules and using them effectively. This exercise employs reasoning to handle absolute value, considering two opposite cases:

• If \(2 - 3x\) is a positive value that equals 13, find \(x\).
• If \(2 - 3x\) is negative and equals -13, find \(x\).

By looking at both cases, we make sure no potential solutions for \(x\) are missed. Mathematical reasoning ensures that both scenarios are considered, providing a comprehensive answer for the function \(y = |2 - 3x|\). This reasoning allows us to deduce that a valid mathematical solution should account for every potential path to equality.